Question

A particle of mass m is executing oscillation about origin on x - axis. Its potential energy is $U\left(x\right)=K|X{|}^n$. If

the time period T is function of its mass, amplitude(a) and k; find the value of n for $T\alpha a^{-\frac{1}{2}}$.

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Answer 1

$\left[K\right]=\frac{\left[U\right]}{[X{]}^n}=\frac{\left[M{L}^2{T}^{-2}\right]}{\left[L^n\right]}=\left[M{L}^{2-n}{T}^{-2}\right]$

$T\propto \left(mass{)}^X\right(amp{)}^Y(K{)}^Z$

$\left[M^0{L}^{0}T\right]=\left[M{]}^X\right[L{]}^Y[M{L}^{2-n}{T}^{-2}{]}^Z$

= $\left[M^{X+Z}{L}^{Y+2Z-nZ}{T}^{-2Z}\right]$

So, -2Z = 1 or $Z=-\frac{1}{2}$ ................................(i)

and, X + Z = 0 or $X=-Z=\frac{1}{2}$ ........................(ii)

and, Y + 2Z-nZ = 0 or$Y=1-\frac{n}{2}$[from(i) and (ii)]

So, $T\propto \left(mass{)}^{\frac{1}{2}}\right(amp{)}^{1-\frac{n}{2}}(K{)}^{-\frac{1}{2}}$

Given $T\propto (amp{)}^{-\frac{1}{2}}$

So $1-\frac{n}{2}=-\frac{1}{2}$ $\Rightarrow n=3$